Assuming \(\psi (4415)\)  as the \(\psi (4S)\)  state in the LP model, the calculated decay widths are
listed in Table REF . It shows that the predicted total width, \(\Gamma \simeq 67\)  MeV, is consistent
with the measured value \(62\pm 20\)  MeV [1]}.
The main decay modes are \(DD_2\) , \(DD_1^{\prime }\) , \(DD_1\)  and
\(D^*D_0\)  In this work, \(D_1\)  and \(D_1^{\prime }\)  stand for the narrow state
\(D_1(2420)\)  and broad state \(D_1(2430)\)  listed in the PDG [1]}, respectively,
which are considered to be mixed states via \(^3P_1\) -\(^1P_1\)  mixing as defined in Ref. [3]}.
While \(D_0\)  and \(D_2\)  stand for the states \(D_0(2400)\)  and \(D_2(2460)\)  listed in the PDG [1]}, respectively.. The branching fractions of \(Br[\psi (4415)\rightarrow DD_2, \ DD_1^{\prime }, \ DD_1, \ D^*D_0]\)  are \(\mathcal {O}(10\%)\) .
Our results for some channels are significantly different from those in [5]}.
This difference is mainly due to the different wave functions of initial states.
For example, in the \(\psi (4415) \rightarrow D^* D_0^*\)  case,
if the wave function of initial state \(\psi (4415)\)  in Ref. [5]} is replaced by that of our LP model,
we get a larger partial wave amplitude \(^3S_1 = -0.1426\)  GeV\(^{-1/2}\) 
than \(^3S_1 = -0.00087 \)  GeV\(^{-1/2}\)  obtained in Ref. [5]}.
This reflects the fact that the integral part \(I^{M_{L_A},m}_{M_{L_B},M_{L_C}}(\mathbf {P})\)  is sensitive to
the node position of the wave function of initial states.
